Chapter 1

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. You are choosing between two long-distance telephone plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per minute for all long- distance calls. Plan B has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per minute for all long-distance calls. How many minutes of long-distance calls in a month make plan A the better deal?

use a graphing utility and the graph's \(x\)-intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$ \begin{aligned} &\sqrt{2 x+13}-x-5=0\\\ &[-5,5,1] \text { by }[-5,5,1] \end{aligned} $$

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

The formula $$ N=2 x^{2}+22 x+320 $$ models the number of inmates, \(N,\) in thousands, in U.S. state and federal prisons \(x\) years after 1980 . The graph of the formula is shown in a [0,20,1] by [0,1600,100] viewing rectangle at the top of the next column. Use the formula to solve Exercises 105-106. (graph cant copy) In which year were there 740 thousand inmates in U.S. state and federal prisons? Identify the solution as a point on the graph shown.

Which one of the following is true? a. Squaring both sides of \(\sqrt{y+4}+\sqrt{y-1}=5\) leads to \(y+4+y-1=25,\) an equation with no radicals. b. The equation \(\left(x^{2}-2 x\right)^{9}-5\left(x^{2}-2 x\right)^{3}+6=0\) is quadratic in form and should be solved by letting \(t=\left(x^{2}-2 x\right)^{3}\) c. If a radical equation has two proposed solutions and one of these values is not a solution, the other value is also not a solution. d. None of these statements is true.

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it cost \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to have a profit gain?

The formula $$ N=2 x^{2}+22 x+320 $$ models the number of inmates, \(N,\) in thousands, in U.S. state and federal prisons \(x\) years after 1980 . The graph of the formula is shown in a [0,20,1] by [0,1600,100] viewing rectangle at the top of the next column. Use the formula to solve Exercises 105-106. (graph cant copy) In which year were 1100 thousand inmates in U.S. state and federal prisons? Identify the solution as a point on the graph shown?

Solve: \(\sqrt{6 x-2}=\sqrt{2 x+3}-\sqrt{4 x-1}\)

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it cost \(\$ 3.00\) to produce each package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to have a profit gain?

A baseball diamond is actually a square with 90 -foot sides. What is the distance from home plate to second base? (Image cant copy)

Solve without squaring both sides: $$ 5-\frac{2}{x}=\sqrt{5-\frac{2}{x}} $$

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

A 20 -foot ladder is 15 feet from the house. How far up the house does the ladder reach? (Image cant copy)

Solve for \(x: \sqrt[3]{x \sqrt{x}}=9\)

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

An 8 -foot tree is supported by two wires that extend from the top of the tree to a point on the ground located 15 feet from the base of the tree. Find the total length of the two support wires.

Solve for \(x: \quad x^{5 / 6}+x^{2 / 3}-2 x^{1 / 2}=0\)

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. On two examinations, you have grades of 86 and \(88 .\) There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90 a. What must you get on the final to earn an A in the course? b. By taking the final, if you do poorly, you might risk the \(\mathrm{B}\) that you have in the course based on the first two exam grades. If your final average is less than 80 , you will lose your \(\mathbf{B}\) in the course. Describe the grades on the final that will cause this happen.

A vertical pole is supported by three wires. Each wire is 13 yards long and is anchored 5 yards from the base of he pole. How far up the pole will the wires be attached?

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. Parts for an automobile repair cost \(\$ 175 .\) The mechanic charges \(\$ 34\) per hour. If you receive an estimate for at least \(\$ 226\) and at most \(\$ 294\) for fixing the car, what is the time interval that the mechanic will be working on the job?

The length of a rectangular garden is 5 feet greater than the width. The area of the garden is 300 square feet. Find the length and the width.

When graphing the solutions of an inequality, what does a parenthesis signify? What does a bracket signify?

A rectangular parking lot has a length that is 3 yards greater than the width. The area of the rectangular lot is 180 square yards. Find the length and the width.

When solving an inequality, when is it necessary to change the sense of the inequality? Give an example.

A machine produces open boxes using square sheets of metal. The figure illustrates that the machine cuts equal-sized squares measuring 2 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 200 cubic inches, find the length of the side of the open square-bottom box.

Describe ways in which solving a linear inequality is similar to solving a linear equation.

A machine produces open boxes using square sheets of metal. The machine cuts equal-sized squares measuring 3 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 75 cubic inches, find the length of the side of the open square-bottom box.

Describe ways in which solving a linear inequality is different than solving a linear equation.

What is a compound inequality and how is it solved?

A piece of wire is 8 inches long. The wire is cut into two pieces and then each piece is bent into a square. Find the length of each piece if the sum of the areas of these squares is to be 2 square inches. (image cant copy)

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

A painting measuring 10 inches by 16 inches is surrounded by a frame of uniform width. If the combined area of the painting and the frame is 280 square inches, determine the width of the frame.

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

What is a quadratic equation?

Explain why \(|x|<-4\) has no solution.

Explain how to solve x^{2}+6 x+8=0 using factoring and the zero-product principle.

Describe the solution set of \(|x|>-4\).

Explain how to solve x^{2}+6 x+8=0 by completing the square.

The formula $$V=3.5 x+120$$ models Super Bowl viewers, \(V,\) in millions, \(x\) years after \(1990 .\) Use the formula to write a word problem that can be solved using a linear inequality. Then solve the problem.

Explain how to solve x^{2}+6 x+8=0 using the quadratic formula.

In Exercises 121-122, solve each inequality using a graphing utility. Graph each side separately. Then determine the values of \(x\) for which the graph on the left side lies above the graph on the right side. $$-3(x-6)>2 x-2$$

How is the quadratic formula derived?

Solve each inequality using a graphing utility. Graph each side separately. Then determine the values of \(x\) for which the graph on the left side lies above the graph on the right side. $$-2(x+4)>6 x+16$$

What is the discriminant and what information does it provide about a quadratic equation?

Use the same technique employed in Exercises 121-122 to solve each inequality in Exercises 123-124. In each case, what conclusion can you draw? What happens if you try solving the inequalities algebraically? $$12 x-10>2(x-4)+10 x$$

If you are given a quadratic equation, how do you determine which method to use to solve it?

Use the same technique employed to solve each inequality. In each case, what conclusion can you draw? What happens if you try solving the inequalities algebraically? $$2 x+3>3(2 x-4)-4 x$$

If (x+2)(x-4)=0 indicates that x+2=0 or x-4=0, explain why (x+2)(x-4)=6 does not mean x+2=6 or x-4=6 . Could we solve the equation using x+2=3 and x-4=2 because 3 \cdot 2=6?

Which one of the following is true? a. The first step in solving \(|2 x-3|>-7\) is to rewrite the inequality as \(2 x-3>-7\) or \(2 x-3<7\) b. The smallest real number in the solution set of \(2 x>6\) is 4 c. All irrational numbers satisfy \(|x-4|>0\) d. None of these statements is true.

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) \(2>1 \quad\) This is a true statement. \(2(y-x)>1(y-x) \quad\) Multiply both sides by \(y-x\) \(2 y-2 x>y-x \quad\) Use the distributive property. \(y-2 x>-x \quad\) Subtract \(y\) from both sides. \(y>x quad\) Add \(2 x\) to both sides. The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\).